How to decorate cyclinder
I write this as an answer to be able to add a picture of what I guessed you want: is it something like this?



I can explain step-by-step how to get it, but I wish to be sure that I did understand well...

this was obtained starting from a simple cylinder and then an image like this:



Let me know, in comments, if you wished to get something like this, and if you still need instructions, I'll add to this answer...

[edit] Ok, since you still need it I try a complete step by step description of how I got that vase above. I hope your hospital trip was nothing too bad, anyway :)

The overall descriptio is this: we're going to create a quite detailed vase, first, applying all modifiers, in order to obtain a good-looging basic shape, with smooth geometry (remember that 3d printers don't smooth geometry as 3d programs do, so to get printed smooth results, your mesh needs to be really smooth, and thus, detailed).

Then, we will duplicate a rectangular portion of the outside conical surface of the vase, that we will use to create an "extruded" flower shape, using a displacement modifier. This extrusion needs to be far more detailed than the vase, because this is required to have a good displacement result, but the displacement will be applied to a curved surface, so the "extruded" flower shape will nicely follow the overall vase curvature.

Done that, you'll have two separate shapes: the basic vase, and a rectangular portion of its outside surface dispalced as a fower: you can then join the two objects, and try to export this to STL format, and print that: in my experience, 3d printer's software (like CURA or others) can print an object if they can spot a "solid" shape, even if it comes from a roughly joined mesh,like in this case.

Here's how to:


First step is start from a basic cylinder, remove top face, and scale lower face, to get a basic vase shape, giving it a "brick red" material (cyles renderer, later on this) 





Then, use a solidify modifier to give some depth to your vase mesh, and apply it whan satisfied





Then, since we're going to use a subdivision modifier (subsurf), let's add some extra edge loop near the top and bottom of the vase shape (and also on the top ring of the vase), otherwise it would distort too much there. You can do this with CTRL+R command: a new purple edge appears, then move it where you need, then apply (right click or enter).





Now let's apply the subsurf modifier, level 2 is enough, 





and then apply it, getting a quite dense and smooth basic shape





Now move the view to front, and select a rectangular shape of faces





Then duplicate it





And then separate this duplicate part, obtaining a separate object. Don't move, scale rotate this separate object or the vase, they need to stay where they are now! 





To get a better view of the separate object, hide the vase (select it and press H)





Now you could make this object mesh more evenly spaced, but preserving its curvature: I'm not sure this is really needed, but I did so, dissolving some edges





...ending with just this central one





then, using again CTRL+R  I subdivided the upper and lower parts, ending with the same amount of vertical and orizontal loops. 





then I subdivided this mesh 20 times (needed for later displacement): you'll get a VERY dense mesh...





then I loaded a black flower on white background image in the image/UV editor, like this. The flower image is also set as the texture of the vase "brick red" material (see texture settings)





in edit mode, using U on the complete selected mesh, I chose "project from view (bounds)" to get a nice fill of the UV grid on the image (on the left)





Then, move to modifiers and insert a displacement one, select a small negative strength, select "UV" texture coordinates, and set the object UV map as "UV map"





What you'll get in object mode is like this:





Now unhide the vase you hidden before since it's invisible now, see the outliner, and clik on the disabled "eye" icon to make it visible again, you'll get something like this:





Place camera and lights in a good way to see the result, which should be something like the above result:


 

I know this isn't too short, but is not that complex, after all. Let me know if some clarification is needed. Meanwhile, I thought of another method, maybe simpler, to get similar results... If it works, I'll add another answer, or add to this one.